A-invariant Subspace Research Articles

An A-invariant subspace for bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.

Real and complex invariant subspaces for matrices which are H-positive real in an indefinite inner product space

In this paper, the equivalence of the existence of unique real and complex A-invariant semidefinite subspaces for real H-positive real matrices are shown

Description of characteristic non-hyperinvariant subspaces over the field [formula omitted

Given a square matrix A, an A-invariant subspace is called hyperinvariant (respectively, characteristic) if and only if it is also invariant for all matrices T (respectively, nonsingular matrices T) that commute with A. Shodaʼs Theorem gives a necessary and sufficient condition for the existence of characteristic non-hyperinvariant subspaces for a nilpotent matrix in GF(2). Here we present an explicit construction for all subspaces of this type.

On invariant subspaces of matrices: A new proof of a theorem of Halmos

lmos proved that if A is a matrix and if E is an A-invariant subspace, then there exist matrices B and C such that BA = AB , CA = AC , E is the kernel of B and E is the range of C. We present an elementary proof of this result and show that there exist B and C that additionally satisfy BC = CB = O .

Mathematical systems theory (third edition), G. J. Olsder and J. W. van der Woude, VSSD, Delft, The Netherlands, 2005, 208pp., ISBN 90‐71301‐40‐0

The book Mathematical Systems Theory by G. J. Olsder and J. W. van der Woude is essentially based on courses given by the authors at Delft University during more than 20 years at an undergraduate level. This is the third edition of these lecture notes which is now available. As a result it is a very good reference book, which reflects the feedback of students and benefits from a number of improvements and corrections compared to the previous editions. Of course, the content of this book, mainly centred on the theory of linear time invariant systems, leads to define it as a classical book. This also leads to put it in a category where a huge amount of literature is already available. The main question is then inevitably what is the usefulness of such a new reference for the automatic control community? The book is short, about 200 pages, which is in my opinion a major quality for a text book. Although presented as a mathematical book, it does not contain long and technical preliminaries in which we usually do not know what is really important to be understood. Here, the necessary mathematical tools are introduced just when needed and with just the level of abstraction which is needed. The book contains a chapter on modelling principles, with a lot of classical examples from electrical and mechanical engineering and also some applications in other fields. These examples and some others appear further in the book to illustrate new concepts or new results. Each chapter is ended by an important set of exercises which a good mix between direct numerical application of the results of the chapter and questions with a theoretical flavour in the style ‘Show that…’. These two types of exercises are important in the complete understanding of a topic, but they are not so often encountered together in classical references. In the same spirit a chapter is devoted to some MATLAB exercises with solutions. A brief description of the content of the book follows. The book begins by a short introduction chapter which introduces the ideas of system theory and control and a brief history. The Chapter 2 is dedicated to modelling principles with application in various fields. Chapters 3–5 treat in detail (with, in general, the proof of the theorems) the standard aspects of linear systems in state-space representation. Chapter 3 which considers the solution of the state equation begins by some necessary considerations on the linearization problem. Chapter 4 deals with the classical properties of linear systems: stability, controllability and observability but with some attempt to see them with a ‘modern’ or geometric point of view, in particular by using the notion of A-invariant subspace. Chapter 5 is devoted to state and output feedback, pole placement until the separation theorem. Chapter 6 is on the transfer matrix approach, it contains the main results on this point of view but is treated with less details than the previous chapters. Chapter 7 presents rapidly the discrete-time state-space models. Chapter 8 which is called ‘Extensions and some related topics’ proposes some further directions of studies as the behavioural models, polynomial matrix representations, nonlinear, distributed parameter systems and discrete event systems. Some pages are also devoted to optimal control and filtering. This book is mainly written for an undergraduate course on system theory for a department of mathematics, but is also suitable for an electrical engineering audience. Both types of students will find in it familiar subjects and ways of thinking and also more exotic aspects and interests. This book may also be useful for graduate students and researchers whose main field is not in control (say from mechanics, chemistry, biology, etc.). They will find in it a rapid overview of the classical results in control theory, without too difficult mathematics but with a real rigour of presentation. Compared with well-known and excellent references 1-3 it is certainly much less encyclopedic. Although the titles are very similar it is far from the level of mathematical abstraction which may be found in Reference 1 or even in Reference 4. For this book the usual assertion ‘it is readable with a basic knowledge in linear algebra and differential equations’ is really true. It is also certainly much less engineering oriented as References 2, 3. In conclusion, this book may find a very useful place in the literature on linear system theory because of its qualities of presentation and writing, its concision and its good trade-off between rigour and accessibility. It can be considered as an appealing entrance in system theory which will encourage students, and also new researchers in the field, to pursue by the study of more complete books as, for example, those cited before.

Convergence conditions for a restarted GMRES method augmented with eigenspaces

We consider the GMRES(m,k) method for the solution of linear systems Ax=b, i.e. the restarted GMRES with restart m where to the standard Krylov subspace of dimension m the other subspace of dimension k is added, resulting in an augmented Krylov subspace. This additional subspace approximates usually an A-invariant subspace. The eigenspaces associated with the eigenvalues closest to zero are commonly used, as those are thought to hinder convergence the most. The behaviour of residual bounds is described for various situations which can arise during the GMRES(m,k) process. The obtained estimates for the norm of the residual vector suggest sufficient conditions for convergence of GMRES(m,k) and illustrate that these augmentation techniques can remove stagnation of GMRES(m) in many cases. All estimates are independent of the choice of an initial approximation. Conclusions and remarks assessing numerically the quality of proposed bounds conclude the paper. Copyright © 2004 John Wiley & Sons, Ltd.

Semi-bounded restrictions of Dirac type operators and the unique continuation property

Let M be a connected Riemannian manifold and let D be a Dirac type operator acting on smooth compactly supported sections in a Hermitian vector bundle over M. Suppose D has a self-adjoint extension A in the Hilbert space of square-integrable sections. We show that any L 2-section ϕ contained in a closed A-invariant subspace on which the restriction of A is semi-bounded has the unique continuation property: if ϕ vanishes on a non-empty open subset of M, then it vanishes on all of M

Invariant subspaces and representations of certain von Neumann algebras

Let (N,α,G) be a covariant system and let (π, U) be a covariant representation of (N,α,G) on a Hilbert space H. In this note, we investigate the representation of the covariance algebra M and the σ-weakly closed subalgebra A generated by π(N) and {Ug}g≥0 in the case of G = Z or R when there exists a pure, full, A-invariant subspace of H.

On the orbit of invariant subspaces of linear operators in finite-dimensional spaces (new proof of a Halmos's result)

P.R. Halmos proved that for a linear operator A over a finite-dimensional complex vector space E, every A-invariant subspace of E is the range of a commutant of A. His proof was based on a generalization of the concept of eigenvector. In this note, we give an invariant proof of this Halmos's theorem.

Stability of Linear Difference and Differential Inclusions

Any given i>n×i>n matrix i>A is shown to be a restriction, to the i>A-invariant subspace, of a nonnegative i>N×i>N matrix i>B of spectral radius ρ(i>B) arbitrarily close to ρ(i>A). A difference inclusion x^{k+1}\in\mathbb{A} x^{k}, where \mathbb{A} is a compact set of matrices, is asymptotically stable if and only if \mathbb{A} can be extended to a set \mathbb{B} of nonnegative matrices i>B with \|B\|_{1}<1 or \|B\|_{\infty}<1. Similar results are derived for differential inclusions.

On Pták's derivation of the Jordan normal form

Some readers of [1] might appreciate the following comments that make more explicit how Ptak’s beautiful insight there leads to a trivial proof of (the basics of) the Jordan normal form. The proof of Theorem 1 of [1] can also be based on the observation that, X being finite-dimensional, the sequence {0} ⊆ ker A ⊆ ker A2 ⊆ · · · must eventually be stationary, i.e., ker A = ker Aq+p for some q and all p > 0. For such q, let Xr and Xs be the range and the kernel, respectively, of A , hence dim X = dim Xr + dim Xs . Further, for any x ∈ Xr ∩ Xs, x = Az for some z, and so z ∈ ker A2q = ker A , hence x = 0. Therefore, X is the direct sum of the two A-invariant subspaces Xs and Xr , and A is regular on Xr (since A is) and nilpotent on Xs . In the setup and notation of Theorem 2 of [1], there must be, by duality, some y0 in Y for which 〈x0Aq−1, y0〉 / = 0, hence the q-order matrix (〈x0Aj−1, y0A∗q−i〉: i,j = 1, . . . , q) is triangular with nonzero diagonal entries, therefore invertible, and this guarantees that X is the direct sum X0 + X′, with X0 the linear span of (x0Aj−1: j = 1, . . . , q) and X′ the annihilator of {y0A∗q−i : i = 1, . . . , q}, both of which are Ainvariant. Moreover, it shows (x0Aj−1: j = 1, . . . , q) to be a basis for X0, and the matrix representation, with respect to this basis, of A restricted to X0 has the familiar form of a Jordan block (for the eigenvalue 0). Now, X being finite-dimensional, there are A-invariant direct sum decompositions X = X1 + · · · + Xm that are minimal in the sense that none of its summands is the direct sum of two nontrivial A-invariant subspaces. Take any one such. Then the matrix representation for A with respect to any basis made up from bases for the summands Xi is block diagonal, with the ith block the matrix representation of the restriction Ai of A to Xi with respect to the chosen basis for Xi . Assuming the underlying field to be algebraically closed, the restriction Ai of A to Xi has some eigenvalue, λi , and, in view of the minimality of Xi , Theorem 1 ensures

Orbits of invariant subspaces of algebraic linear operators

Let E be a vector space over a field K, and A an algebraic K-linear operator on E. We describe the orbits under the action of the commutant of A and of the group of the invertible elements of the commutant of A on the lattice of A-invariant subspaces.

Lattice properties of sets of semidefinite solutions of continuous-time algebraic riccati equations

The set of negative-semidefinite solutions of the algebraic Riccati equation A∗X + XA + XBB∗X − C∗C = 0 is studied under the most general hypothesis, namely that (A, B) is stabilizable modulo the undetectable subspace of (A, C). If is a lattice, then is isomorphic to 0 ⊕ S where T0 is the direct sum of copies of the set of nonnegative real numbers and is a closure of which is order isomorphic to a system of well-defined A-invariant subspaces. Necessary and sufficient conditions are given such that (or) is a lattice.

Decomposition and Parametrization of Semidefinite Solutions of the Continuous-Time Algebraic Riccati Equation

Negative-semidefinite solutions of the $\operatorname{ARE}\mathcal{R}(X) = A^ * X + XA + XBB^ * X - C^ * C = 0$ are studied. With respect to an appropriate basis the ARE breaks up into a Lyapunov equation $A_0^ * X_0 + X_0 A_0 = 0$, where $A_0 $ has only purely imaginary eigenvalues, and an indecomposable Riccati equation $\mathcal{R}_r (X_r ) = A_r^ * X_r + X_r A_r + X_r B_r B_r^ * X_r - C_r^ * C_r = 0$ such that each solution $x \leq 0$ is of the form $X = {\operatorname {diag}}(X_0 ,X_r )$ . The focus is on the solutions $\mathcal{S} = \{ X| X . = {\operatorname {diag}}(0,X_r ),\mathcal{R}_r (X_r ) = 0,X_r \} \leq 0$. The set $\mathcal{S}$ has as an order-isomorphic image a well-defined set $\mathcal{N}$ of A-invariant subspaces. The characterization of $\mathcal{N}$ involves the stabilizable and the uncontrollable subspace of $(A,B,C)$.

Invariant Neutral Subspaces for Symmetric and Skew Real Matrix Pairs

AbstractReal matrix pairs (A,H) satisfying det H ≠ 0, HT = εH, and HA - ηATH, where ε, η take the values +1 or —1, are considered. It is shown that maximal A-invariant H-neutral subspaces have the same dimension (depending on ε and η), called the order of neutrality of the pair (A, H). The order of neutrality of definitizable pairs is investigated. In particular, this concept is used to obtain lower bounds for the number of pure imaginary eigenvalues of low rank perturbations of definitizable pairs when (ε,η) = (1, - 1 ) and when (ε,η) = (—1,—1).

Stability index of invariant subspaced of matrices

Let A be an n × n nonderogatory matrix. Given an A-invariant subspace M, we identify the best possible rate of convergence Mm →M, where for =1,2,…M m is a Bm -invariant subspace, and is any sequence of matrices converging to A. Thereby the stability of a stable invariant subspace is cimpletely described (the problems concerning the stability index have been raised in [7]).

Extensions of Jordan bases for invariant subspaces of a matrix

A characterization is obtained for the matrices A with the property that every (some) Jordan basis of every A-invariant subspace can be extended to a Jordan basis of A. These results are based on a criterion for a Jordan basis of an invariant subspace to be extendable to a Jordan basis of the whole space. The criterion involves two concepts: the constancy property and the depth property.

On strictly positively invariant cones

For a matrix A ∈ R n × n , it is shown that strict positive invariance of a proper cone C ⊂ R n (that is, e tA C/{0} ⊂ int C ∀ t 0) implies the existence of a certain direct sum decomposition of R n into A-invariant subspaces. Our results lead to a characterization of the set of initial points which give rise to solution curves that reach S , under the differential equation ẋ = Ax. Also given is an application in stability theory.

On commuting nilpotent matrices

Let F be a field, n a positive integer, A a nilpotent n x n-matrix and V the vector space of n-tuples over F. A operates on V in a natural way. k Let V = ~ V~ be a decomposition of V, where the V~ are A-invariant i=1 and A-indecomposable for i = 1 . . . . . k, and dim V~ > dim V~+I for i = 1 . . . . . k 1 . The k-tUple of positive integers (al . . . . , a k ) : = (dim 1/1 . . . . . dim Vk) is called the Segre-type of A (see [3], p. 48). We will also say that the A-space V is of type ( a l , . . . , ak). Let Wbe an A-invariant subspace of V. A induces nilpotent linear transformations on W and V/W. It is known that we can assign to these linear transformations Segre-types (bl . . . . . bk) and (cl . . . . . Ck) with b~-> bi+l and c~ >_ c~+1 for i = 1 . . . . , k 1 ; the bi and c~ are non-negative integers with the property b~ < as and c~ < as for all i = 1 , . . . , k. We will also say that the A-spaces Wand V / W are of type (bt . . . . . bk) and (cl , . . . , Ck) respectively. The following lemma gives a criterion for Wto be C(A)-invariant, where C(A) is the ring of all n x n-matrices which commute with A.